# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020023

## Quaternary group ring codes: Ranks, kernels and self-dual codes

 1 Department of Mathematics, University of Scranton, Scranton, PA 18510, USA 2 Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain 3 Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86001, USA

Received  July 2018 Published  September 2019

Fund Project: This work has been partially supported by the Spanish MINECO under Grant TIN2016-77918-P (AEI/FEDER, UE)

We study $G$-codes over the ring ${\mathbb{Z}}_4$, which are codes that are held invariant by the action of an arbitrary group $G$. We view these codes as ideals in a group ring and we study the rank and kernel of these codes. We use the rank and kernel to study the image of these codes under the Gray map. We study the specific case when the group is the dihedral group and the dicyclic group. Finally, we study quaternary self-dual dihedral and dicyclic codes, tabulating the many good self-dual quaternary codes obtained via these constructions, including the octacode.

Citation: Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2020023
##### References:
 [1] E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type Ⅱ codes, even unimodular lattices, and invariant rings, IEEE Trans. Inform. Theory, 45 (1999), 1194-1205. doi: 10.1109/18.761269. Google Scholar [2] W. Bosma, J. J. Cannon and C. Fieker, A. Steel: Handbook of Magma functions, Edition 2.22 5669 pages, 2016, http://magma.maths.usyd.edu.au/magma/.Google Scholar [3] J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo $4$, J. Combin. Theory Ser. A, 62 (1993), 30-45. doi: 10.1016/0097-3165(93)90070-O. Google Scholar [4] S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2. Google Scholar [5] S. T. Dougherty and C. Fernández-Córdoba, Codes over $\Bbb Z_{2^k}$, Gray maps and self-dual codes, Adv. in Math. of Commun., 5 (2011), 571-588. doi: 10.3934/amc.2011.5.571. Google Scholar [6] S. T. Dougherty and C. Fernández-Córdoba, Kernels and ranks of cyclic and negacyclic quaternary codes, Des. Codes Cryptogr., 81 (2016), 347-364. doi: 10.1007/s10623-015-0163-6. Google Scholar [7] S. T. Dougherty, C. Fernández-Córdoba and R. Ten-Valls, Quasi-cyclic codes as cyclic codes over a family of local rings, Finite Fields Appl., 40 (2016), 138-149. doi: 10.1016/j.ffa.2016.04.002. Google Scholar [8] S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak, Group Rings, $G$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138. doi: 10.1007/s10623-017-0440-7. Google Scholar [9] C. Fernández-Córdoba, J. Pujol and M. Villanueva, On rank and kernel of ${\mathbb{Z}}_4$-linear codes, Lecture Notes in Computer Science, 5228 (2008), 46-55. Google Scholar [10] R. A. Ferraz, F. S. Dutra and C. Polcino Milies, Semisimple group codes and dihedral codes, Algebra Discrete Math., (2009), 28-48. Google Scholar [11] J. Gildea, A. Kaya, R. Taylor and B. Yildiz, Constructions for self-dual codes induced from group rings, Finite Fields Appl., 51 (2018), 71-92. doi: 10.1016/j.ffa.2018.01.002. Google Scholar [12] M. Guerreiro, Group algebras and coding theory, São Paulo Journal of Mathematical Sciences, 10 (2016), 346-371. doi: 10.1007/s40863-016-0040-x. Google Scholar [13] A. R. J. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The ${\mathbb{Z}}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154. Google Scholar [14] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math., 31 (2006), 319-33. Google Scholar [15] F. J. MacWilliams, Binary codes which are ideals in the group algebra of an Abelian group, Bell System Tech. J., 49 (1970), 987-1011. doi: 10.1002/j.1538-7305.1970.tb01812.x. Google Scholar [16] J. MacWilliams, Codes and ideals in group algebras, Combinatorial Mathematics and its Applications, Univ. North Carolina Press, Chapel Hill, N.C., (1969), 317–328. Google Scholar [17] O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508. doi: 10.2307/1968173. Google Scholar [18] V. Pless, P. Solé and Z. Q. Qian, Cyclic self-dual ${\mathbb{Z}}_4$-codes, with an appendix by Pieter Moree, Finite Fields Appl., 3 (1997), 48-69. doi: 10.1006/ffta.1996.0172. Google Scholar [19] V. S. Pless and Z. Q. Qian, Cyclic codes and quadratic residue codes over ${\mathbb{Z}}_4$, IEEE Trans. Inform. Theory, 42 (1996), 1594-1600. doi: 10.1109/18.532906. Google Scholar [20] D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, 80, Springer-Verlag, New York, 1993. Google Scholar

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##### References:
 [1] E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type Ⅱ codes, even unimodular lattices, and invariant rings, IEEE Trans. Inform. Theory, 45 (1999), 1194-1205. doi: 10.1109/18.761269. Google Scholar [2] W. Bosma, J. J. Cannon and C. Fieker, A. Steel: Handbook of Magma functions, Edition 2.22 5669 pages, 2016, http://magma.maths.usyd.edu.au/magma/.Google Scholar [3] J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo $4$, J. Combin. Theory Ser. A, 62 (1993), 30-45. doi: 10.1016/0097-3165(93)90070-O. Google Scholar [4] S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2. Google Scholar [5] S. T. Dougherty and C. Fernández-Córdoba, Codes over $\Bbb Z_{2^k}$, Gray maps and self-dual codes, Adv. in Math. of Commun., 5 (2011), 571-588. doi: 10.3934/amc.2011.5.571. Google Scholar [6] S. T. Dougherty and C. Fernández-Córdoba, Kernels and ranks of cyclic and negacyclic quaternary codes, Des. Codes Cryptogr., 81 (2016), 347-364. doi: 10.1007/s10623-015-0163-6. Google Scholar [7] S. T. Dougherty, C. Fernández-Córdoba and R. Ten-Valls, Quasi-cyclic codes as cyclic codes over a family of local rings, Finite Fields Appl., 40 (2016), 138-149. doi: 10.1016/j.ffa.2016.04.002. Google Scholar [8] S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak, Group Rings, $G$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138. doi: 10.1007/s10623-017-0440-7. Google Scholar [9] C. Fernández-Córdoba, J. Pujol and M. Villanueva, On rank and kernel of ${\mathbb{Z}}_4$-linear codes, Lecture Notes in Computer Science, 5228 (2008), 46-55. Google Scholar [10] R. A. Ferraz, F. S. Dutra and C. Polcino Milies, Semisimple group codes and dihedral codes, Algebra Discrete Math., (2009), 28-48. Google Scholar [11] J. Gildea, A. Kaya, R. Taylor and B. Yildiz, Constructions for self-dual codes induced from group rings, Finite Fields Appl., 51 (2018), 71-92. doi: 10.1016/j.ffa.2018.01.002. Google Scholar [12] M. Guerreiro, Group algebras and coding theory, São Paulo Journal of Mathematical Sciences, 10 (2016), 346-371. doi: 10.1007/s40863-016-0040-x. Google Scholar [13] A. R. J. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The ${\mathbb{Z}}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154. Google Scholar [14] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math., 31 (2006), 319-33. Google Scholar [15] F. J. MacWilliams, Binary codes which are ideals in the group algebra of an Abelian group, Bell System Tech. J., 49 (1970), 987-1011. doi: 10.1002/j.1538-7305.1970.tb01812.x. Google Scholar [16] J. MacWilliams, Codes and ideals in group algebras, Combinatorial Mathematics and its Applications, Univ. North Carolina Press, Chapel Hill, N.C., (1969), 317–328. Google Scholar [17] O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508. doi: 10.2307/1968173. Google Scholar [18] V. Pless, P. Solé and Z. Q. Qian, Cyclic self-dual ${\mathbb{Z}}_4$-codes, with an appendix by Pieter Moree, Finite Fields Appl., 3 (1997), 48-69. doi: 10.1006/ffta.1996.0172. Google Scholar [19] V. S. Pless and Z. Q. Qian, Cyclic codes and quadratic residue codes over ${\mathbb{Z}}_4$, IEEE Trans. Inform. Theory, 42 (1996), 1594-1600. doi: 10.1109/18.532906. Google Scholar [20] D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, 80, Springer-Verlag, New York, 1993. Google Scholar
(Extremal) Dihedral Self-dual Codes of length 4
 First Row of A First row of B Min Lee Weight Lee Weight Distribution (1, 1) (1, 3) 4 $1+14z^4+z^{8}$ (1, 1) (3, 1) 4 $1+14z^4+z^{8}$ (1, 3) (1, 1) 4 $1+14z^4+z^{8}$ (1, 3) (3, 3) 4 $1+14z^4+z^{8}$ (3, 1) (1, 1) 4 $1+14z^4+z^{8}$ (3, 1) (3, 3) 4 $1+14z^4+z^{8}$ (1, 3) (1, 3) 4 $1+14z^4+z^{8}$ (3, 3) (3, 1) 4 $1+14z^4+z^{8}$
 First Row of A First row of B Min Lee Weight Lee Weight Distribution (1, 1) (1, 3) 4 $1+14z^4+z^{8}$ (1, 1) (3, 1) 4 $1+14z^4+z^{8}$ (1, 3) (1, 1) 4 $1+14z^4+z^{8}$ (1, 3) (3, 3) 4 $1+14z^4+z^{8}$ (3, 1) (1, 1) 4 $1+14z^4+z^{8}$ (3, 1) (3, 3) 4 $1+14z^4+z^{8}$ (1, 3) (1, 3) 4 $1+14z^4+z^{8}$ (3, 3) (3, 1) 4 $1+14z^4+z^{8}$
(Extremal) Dihedral Self-dual Codes of length 8
 First Row of $A$ First row of $B$ Min Lee Weight Lee Weight Distribution (0, 0, 2, 2) (1, 1, 3, 1) 4 $1+28z^4+198z^{8}+\dots$ (0, 0, 0, 0) (1, 3, 1, 1) 4 $1+28z^4+198z^{8}+\dots$ (0, 0, 0, 2) (3, 1, 3, 1) 4 $1+12z^4+64z^{6}+102z^8+\dots$ (0, 0, 2, 0) (1, 1, 3, 3) 4 $1+12z^4+64z^{6}+102z^8+\dots$
 First Row of $A$ First row of $B$ Min Lee Weight Lee Weight Distribution (0, 0, 2, 2) (1, 1, 3, 1) 4 $1+28z^4+198z^{8}+\dots$ (0, 0, 0, 0) (1, 3, 1, 1) 4 $1+28z^4+198z^{8}+\dots$ (0, 0, 0, 2) (3, 1, 3, 1) 4 $1+12z^4+64z^{6}+102z^8+\dots$ (0, 0, 2, 0) (1, 1, 3, 3) 4 $1+12z^4+64z^{6}+102z^8+\dots$
Best Dicyclic Self-dual Codes of lengths 4, 8 and 12
 $n$ 1st row of $A$ 1st row of $B$ 1st row of $C$ Min Lee Weight Gray Image Linear $4$ (1, 3) (3, 3) (3, 3) 4 Yes $8$ (0, 0, 0, 2) (3, 3, 3, 3) (3, 3, 3, 3) 4 Yes $8$ (0, 0, 1, 1) (0, 0, 1, 3) (1, 3, 0, 0) 4 No $8$ (0, 0, 1, 1) (0, 1, 1, 2) (1, 2, 0, 1) $6^*$ No $12$ (0, 0, 0, 0, 0, 0) (0, 1, 3, 0, 1, 1) (0, 1, 1, 0, 1, 3) 4 Yes
 $n$ 1st row of $A$ 1st row of $B$ 1st row of $C$ Min Lee Weight Gray Image Linear $4$ (1, 3) (3, 3) (3, 3) 4 Yes $8$ (0, 0, 0, 2) (3, 3, 3, 3) (3, 3, 3, 3) 4 Yes $8$ (0, 0, 1, 1) (0, 0, 1, 3) (1, 3, 0, 0) 4 No $8$ (0, 0, 1, 1) (0, 1, 1, 2) (1, 2, 0, 1) $6^*$ No $12$ (0, 0, 0, 0, 0, 0) (0, 1, 3, 0, 1, 1) (0, 1, 1, 0, 1, 3) 4 Yes
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